Correlated random walks with a finite memory range
type: Item
instance of: scholarly article
zbMATH Open document ID: 0992.60095
DOI: 10.1142/S0129183100000791
author: Roger Bidaux
author: Nino Boccara
published in: International Journal of Modern Physics C
publication date: 2002-09-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/adap-org/9904001
Mathematics Subject Classification ID: 60K40
Mathematics Subject Classification ID: 82B41
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 1606519
zbMATH Keywords: correlated random walk
zbMATH Keywords: finite memory range
zbMATH Keywords: Hurst exponent
zbMATH Keywords: crossover
MaRDI profile type: Publication
arXiv ID: adap-org/9904001
cites work: The correlated random walk
cites work: The Gillis–Domb–Fisher correlated random walk
OpenAlex ID: W2065949133
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title: Correlated random walks with a finite memory range
review text: A one-dimensional correlated random walk \((R_n)_{n\geq 0}\) with (finite) memory range \(M\geq 1\) is given by \(R_n=S_1+S_2+\dots+S_n\), where the increments \(S_1\), \(S_2\), \dots\ are defined as follows: for \(k\leq{M}\), the jumps \(S_k\) are i.i.d.\ symmetric Bernoulli random variables with values~\(\pm 1\); for \(k>M\), a coin showing head with probability \(p\in[0,1]\) is tossed; if there was a head, \(S_k\) takes values \(\pm 1\) with equal probabilities, otherwise, \(S_k=f_{M}(S_{k-1},S_{k-2},\dots,S_{k-M})\), where \(f_{M}\) is a Boolean function from \(\{-1,1\}^M\) onto \(\{-1,1\}\). NEWLINENEWLINENEWLINEA particular case with \(M=1\) was introduced by \textit{G. I. Taylor} in 1921 and was studied by \textit{S. Goldstein} [Q. J. Mech. Appl. Math. 4, 129-156 (1951; Zbl 0045.08102)]. In the present paper, the authors use the transfer-matrix formalism to investigate some cases with \(M>1\) subject to zero mean displacement condition \({\mathbb E}R_n=0\). In particular, they give formulae for the variance \({\mathbb E}(R_n)^2\) and discuss the crossover phenomenon from quadratic to linear behaviour of \({\mathbb E}(R_n)^2\).
reviewed by: Ostap Hryniv
description: scientific article; zbMATH DE number 1606519